Classification of topological phases of parafermionic chains with symmetries

Abstract

We study the topological classification of parafermionic chains in the presence of a modified time-reversal symmetry that satisfies ${\mathcal{T}}^{2}=1$. Such chains can be realized in one-dimensional structures embedded in fractionalized two-dimensional states of matter, e.g., at the edges of a fractional quantum spin Hall system, where counterpropagating modes may be gapped either by backscattering or by coupling to a superconductor. In the absence of any additional symmetries, a chain of ${\mathbb{Z}}_{m}$ parafermions can belong to one of several distinct phases. We find that when the modified time-reversal symmetry is imposed, the classification becomes richer. If $m$ is odd, each of the phases splits into two subclasses. We identify the symmetry-protected phase as a Haldane phase that carries a Kramers doublet at each end. When $m$ is even, each phase splits into four subclasses. The origin of this split is in the emergent Majorana fermions associated with even values of $m$. We demonstrate the appearance of such emergent Majorana zero modes in a system where the constituent particles are either fermions or bosons.